Behavior near the boundary of positive solutions of second order parabolic equations. II

Abstract

A boundary backward Harnack inequality is proved for positive solutions of second order parabolic equations in non-divergence form in a bounded cylinder $Q=\Omega \times \left (0,T\right )$ which vanish on $\partial _xQ=\partial \Omega \times \left (0,T\right )$, where $\Omega$ is a bounded Lipschitz domain in $\mathbb {R}^n$. This inequality is applied to the proof of the Hölder continuity of the quotient of two positive solutions vanishing on a portion of $\partial _xQ.$